Verified answer

[tex]\large\underline{\sf{Solution-}}[/tex]

Given expression is

[tex]\sf \: \sqrt{3 + 2 \sqrt{2} } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt{2 + 1 + 2 \sqrt{2} } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt{ {( \sqrt{2} )}^{2} + {(1)}^{2} + 2 \sqrt{2} } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt{ {( \sqrt{2} )}^{2} + {(1)}^{2} + 2 \times \sqrt{2} \times 1} \\ \\ [/tex]

We kmow,

[tex]\boxed{ \bf{ \: {x}^{2} + {y}^{2} + 2xy = {(x + y)}^{2} \: }} \\ \\ [/tex]

So, here

[tex]\sf \: x = \sqrt{2} \\ \\ [/tex]

[tex]\sf \: y = 1 \\ \\ [/tex]

So, on substituting the values in above formula, we get

[tex]\sf \: = \: \sqrt{( \sqrt{2} + 1)^{2} } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt{2} + 1 \\ \\ [/tex]

Hence,

[tex]\sf \: \bf\implies \: \sqrt{3 + 2 \sqrt{2} } = \: \sqrt{2} + 1 \\ \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]

Answer:

√2+1

Step-by-step explanation:

√(3+2√2)

√(1+2+2√2)

√((√2)^2+1^2+2*(√2)(1))

Using identity, a^2+b^2+2ab = (a+b)^2, we have

√(√2+1)^2

[(√2+1)^2]^1/2

√2+1

So, simplified form of √(3+2√2) = √2+1.